Advertisements
Advertisements
Question
If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Advertisements
Solution
Given that: a = cosθ + isinθ
∴ `(1 + a)/(1 - a) = (1 + cos theta + i sin theta)/(1 - cos theta - i sin theta)`
= `(1 + cos theta + i sin theta)/(1 - cos theta - i sin theta) xx (1 - cos theta + i sin theta)/(1 - cos theta + i sin theta)`
= `(1 - cos theta + i sin theta + cos theta - cos^2 theta + i sin theta cos theta + i sin theta - i sin theta cos theta + i^2 sin^2 theta)/((1 - cos theta)^2 - i^2 sin^2 theta)`
= `(1 + i sin theta - cos^2 theta + i sin theta - sin^2 theta)/(1 + cos^2 theta - 2 cos theta + sin^2 theta)`
= `(sin^2 theta + 2i sin theta - sin^2 theta)/(1 + 1 - 2 cos theta)`
= `(2i sin theta)/(2 - 2 cos theta)`
= `(2i sin theta)/(2(1 - cos theta))`
= `(i sin theta)/(1 - cos theta)`
= `(2 sin theta/2 cos theta/2.i)/(2sin^2 theta/2)`
= `cot theta/2 . i`
Hence, `(1 + a)/(1 - a) = icot theta/2`.
APPEARS IN
RELATED QUESTIONS
Express the given complex number in the form a + ib: i–39
Express the given complex number in the form a + ib: 3(7 + i7) + i(7 + i7)
Let z1 = 2 – i, z2 = –2 + i. Find `"Im"(1/(z_1barz_1))`
Evaluate the following:
\[i^{49} + i^{68} + i^{89} + i^{110}\]
Show that 1 + i10 + i20 + i30 is a real number.
Find the value of the following expression:
1+ i2 + i4 + i6 + i8 + ... + i20
Express the following complex number in the standard form a + i b:
\[\frac{3 + 2i}{- 2 + i}\]
If \[\left( \frac{1 + i}{1 - i} \right)^3 - \left( \frac{1 - i}{1 + i} \right)^3 = x + iy\] find (x, y).
Evaluate the following:
\[x^4 + 4 x^3 + 6 x^2 + 4x + 9, \text { when } x = - 1 + i\sqrt{2}\]
Solve the system of equations \[\text { Re }\left( z^2 \right) = 0, \left| z \right| = 2\].
Solve the equation \[\left| z \right| = z + 1 + 2i\].
If n is any positive integer, write the value of \[\frac{i^{4n + 1} - i^{4n - 1}}{2}\].
Find the principal argument of \[\left( 1 + i\sqrt{3} \right)^2\] .
Write the value of \[\arg\left( z \right) + \arg\left( \bar{z} \right)\].
If \[\left| z + 4 \right| \leq 3\], then find the greatest and least values of \[\left| z + 1 \right|\].
Find the real value of a for which \[3 i^3 - 2a i^2 + (1 - a)i + 5\] is real.
Write the argument of \[\left( 1 + i\sqrt{3} \right)\left( 1 + i \right)\left( \cos\theta + i\sin\theta \right)\].
Disclaimer: There is a misprinting in the question. It should be \[\left( 1 + i\sqrt{3} \right)\] instead of \[\left( 1 + \sqrt{3} \right)\].
If z is a non-zero complex number, then \[\left| \frac{\left| z \right|^2}{zz} \right|\] is equal to
If (x + iy)1/3 = a + ib, then \[\frac{x}{a} + \frac{y}{b} =\]
\[(\sqrt{- 2})(\sqrt{- 3})\] is equal to
If \[z = \left( \frac{1 + i}{1 - i} \right)\] then z4 equals
\[\text { If } z = \frac{1}{(1 - i)(2 + 3i)}, \text { than } \left| z \right| =\]
If \[z = \frac{1}{1 - cos\theta - i sin\theta}\] then Re (z) =
If \[x + iy = \frac{3 + 5i}{7 - 6i},\] then y =
If \[z = \frac{1 + 7i}{(2 - i )^2}\] , then
The value of \[\frac{i^{592} + i^{590} + i^{588} + i^{586} + i^{584}}{i^{582} + i^{580} + i^{578} + i^{576} + i^{574}} - 1\] is
If \[z = a + ib\] lies in third quadrant, then \[\frac{\bar{z}}{z}\] also lies in third quadrant if
If the complex number \[z = x + iy\] satisfies the condition \[\left| z + 1 \right| = 1\], then z lies on
Find a and b if abi = 3a − b + 12i
Find a and b if `1/("a" + "ib")` = 3 – 2i
Find a and b if (a + ib) (1 + i) = 2 + i
Evaluate the following : i–888
Match the statements of column A and B.
| Column A | Column B |
| (a) The value of 1 + i2 + i4 + i6 + ... i20 is | (i) purely imaginary complex number |
| (b) The value of `i^(-1097)` is | (ii) purely real complex number |
| (c) Conjugate of 1 + i lies in | (iii) second quadrant |
| (d) `(1 + 2i)/(1 - i)` lies in | (iv) Fourth quadrant |
| (e) If a, b, c ∈ R and b2 – 4ac < 0, then the roots of the equation ax2 + bx + c = 0 are non real (complex) and |
(v) may not occur in conjugate pairs |
| (f) If a, b, c ∈ R and b2 – 4ac > 0, and b2 – 4ac is a perfect square, then the roots of the equation ax2 + bx + c = 0 |
(vi) may occur in conjugate pairs |
If `((1 - i)/(1 + i))^100` = a + ib, then find (a, b).
State True or False for the following:
The order relation is defined on the set of complex numbers.
The real value of θ for which the expression `(1 + i cos theta)/(1 - 2i cos theta)` is a real number is ______.
Show that `(-1+ sqrt(3)i)^3` is a real number.
